1999/01/01 by Dorothee Schueth · 1 citation
Mathematics · Computer Science · #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Isospectral #Mathematics #Pure mathematics #Simply connected space
paper · doi:10.2307/121026
openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Abstract. We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S 4 × S 3 × S 3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar curvature function changes during the deformation. Moreover, the manifolds are in general not isospectral for the Laplace operator on 1-forms. Introduction. Spectral geometry deals with the mutual influences between the geometry of a Riemannian manifold and the spectrum of the associated Laplace operator acting on smooth functions. Until 1964 it was not known whether the spectrum determines the geometry completely. Then J. Milnor constructed the first counterexample, namely, a pair of isospectral, non-isometric flat tori in dimension